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Consider the function $f(x)=12x_{5}+30x_{4}−160x_{3}+3$ $f(x)$ has inflection points at (reading from left to right) $x=D,E$, and $F$ where $D$ is and $E$ is and $F$ is For each of the following intervals, tell whether $f(x)$ is concave up or concave down. $(−∞,D):(D,E):(E,F):(F,∞):1 $

Solution

The inflection point is found by using

Find the second derivative.

Set the second derivative equal to then solve the equation .

Find the points where the second derivative is .

Split into intervals around the points that could potentially be inflection points.

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